On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime

Abstract

Let $({\epsilon }_t{)}_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_T={\sum }_{t=s+1}^{s+T}{\epsilon }_t{\epsilon }_{t-s}^T/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of ${\epsilon }_t$. This paper investigates the limiting behavior of the singular values of $X_T$ under the so-called {\em ultra-dimensional regime} where $p\to \infty$ and $T\to \infty$ in a related way such that $p/T\to 0$. First, we show that the singular value distribution of $X_T$ after a suitable normalization converges to a nonrandom limit $G$ (quarter law) under the forth-moment condition. Second, we establish the convergence of its largest singular value to the right edge of $G$. Both results are derived using the moment method.

32 pages, 2 figures

References in corpus (2)

• Factor modeling for high-dimensional time series: Inference for the number of factors
• Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero

Cited by in corpus (1)

• Moment approach for singular values distribution of a large auto-covariance matrix